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Symmetric differential forms on complete intersection varieties and applications

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In this paper we study the cohomology of the symmetric powers of the cotangent bundle of complete intersection varieties in projective space. We provide an explicit description of some of those cohomology groups in terms of the equations defining the complete intersection. As a first application, we give a new example illustrating the fact that the dimension of the space of holomorphic symmetric differential forms is not deformation invariant. Then, as our main application, we construct varieties with ample cotangent bundle, providing new results towards a conjecture of Debarre.

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Acknowledgments

This work originated during the author’s phd thesis under the supervision of Christophe Mourougane. We thank him very warmly for his guidance, his time and all the discussions we had. We also thank Junjiro Noguchi and Yusaku Tiba for listening through many technical details. We thank Joël Merker for his many encouragements and for all the interest he showed in this work. We also thank Lionel Darondeau for stimulating discussions and for his suggestions about the presentation of this paper.

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  1. IRMA, UMR 7501, 7 rue René-Descartes, 67084, Strasbourg Cedex, France

    Damian Brotbek

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  1. Damian Brotbek
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Correspondence to Damian Brotbek.

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Brotbek, D. Symmetric differential forms on complete intersection varieties and applications. Math. Ann. 366, 417–446 (2016). https://doi.org/10.1007/s00208-015-1332-7

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  • DOI: https://doi.org/10.1007/s00208-015-1332-7

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